What is the probability that a randomly chosen whole number is square free?

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First, we need to understand what we mean by a square free number. It essentially means that a whole number is not divisible by the square of a prime number.
Let’s take the example of 30. From fundamental theorem of arithmetic, we can write it uniquely in terms of prime numbers as

30= 2 × 3 × 5

Since none of the prime numbers are repeated, 30 is square free.
Now suppose we look at number 12.

12= 2 × 2 × 3

In this case the prime number 2 is repeated and hence 12 is not square free.

 The question we have posed in this blog is -what is the probability that a randomly chosen whole number is square free is not very precise.
The reason being that we know there are an infinitely many primes and there is no associated probability distribution. So, for our discussion we will assume that we are talking about a finite set of whole numbers say a million and ask the question what proportion of those are square free. In a more general sense, we can take a big number x and we can ask as we let x tend to infinity what proportion of whole numbers are square free.

We will look at a heuristic proof since a rigorous proof which exists and that will please the serious mathematicians will involve math which we do not want to get into in this blog.

Now if ask the question, what is the probability that a whole number is divisible by 22 (square of a prime number)?

Then it is just 1/22

Then the probability, that a whole number is not divisible by 22  is just 


Similarly, for 32 and so on for other primes.

Assuming that the primes are independent and finite (we know that this is not strictly correct- remember it is a heuristic proof), it follows from Chinese remainder theorem that we can write the probability that the whole number is not divisible by any of the square of the primes as a product



This we can write as 


But we know from our article on Basel problem earlier in the blog that


But since

We have



which is approximately 0.6079271….

This is truly a remarkably beautiful result. What is square of π doing here which is not in context of geometry where we normally encounter it? This has led mathematicians to ask the more fundamental questions in number theory about how often polynomials can take square free value and is an active area of research.

 Reference:
Prof Matt Baker"s Blog article Probability, Primes and Pi, March 2016 

 All inaccuracies in the article are mine. 

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